https://commalg.subwiki.org/w/index.php?action=history&feed=atom&title=Unique_factorization_and_Bezout_iff_principal_ideal 2026-09-06T16:26:21Z Revision history for this page on the wiki MediaWiki 1.41.2 https://commalg.subwiki.org/w/index.php?title=Unique_factorization_and_Bezout_iff_principal_ideal&diff=1902&oldid=prev 2009-01-31T23:10:34Z

<p>New page: ==Statement== The following are equivalent for an <a href="/wiki/Integral_domain" title="Integral domain">integral domain</a>: * It is a <a href="/w/index.php?title=Fact_about::principal_ideal_domain&amp;action=edit&amp;redlink=1" class="new" title="Fact about::principal ideal domain (page does not exist)">fact about::principal ideal domain</a>. * It is both a <a href="/w/index.php?title=Fact_about::unique_factorization_domain&amp;action=edit&amp;redlink=1" class="new" title="Fact about::unique factorization domain (page does not exist)">fact about::unique factorization domain</a> and a ...</p> <p><b>New page</b></p><div>==Statement==<br /> <br /> The following are equivalent for an [[integral domain]]:<br /> <br /> * It is a [[fact about::principal ideal domain]].<br /> * It is both a [[fact about::unique factorization domain]] and a [[fact about::Bezout domain]].<br /> <br /> ==Related facts==<br /> <br /> * [[Unique factorization and Dedekind iff principal ideal]]<br /> * [[Bezout and Dedekind iff principal ideal]]<br /> <br /> ==Facts used==<br /> <br /> # [[uses::PID implies UFD]]<br /> # [[uses::PID implies Bezout]]<br /> # [[uses::Length of irreducible factorization is strictly multiplicatively monotone on unique factorization domain]]<br /> # [[uses::Strictly multiplicatively monotone norm on Bezout domain is a Dedekind-Hasse norm]]<br /> # [[uses::Dedekind-Hasse norm implies principal ideal ring]]<br /> <br /> ==Proof==<br /> <br /> ===Principal ideal domain implies unique factorization and Bezout===<br /> <br /> This follows from facts (1) and (2).<br /> <br /> ===Unique factorization and Bezout implies principal ideal domain===<br /> <br /> This follows by combining facts (3), (4), and (5).<br /> <br /> ==References==<br /> <br /> ===Textbook references===<br /> <br /> * {{booklink-stated|DummitFoote|294|section 8.3 (&#039;&#039;Unique factorization domains&#039;&#039;)}}</div>

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